No-arbitrage condition in some modified HJM framework

Let us consider a forward market, i.e. in which a whole forward curve (different prices corresponding to different maturity dates) is quoted at each time, and assume, for simplicity, that one (Brownian) risk factor causes all market uncertainty.

We are interested in the no-arbitrage condition imposed by the presence of one single source of risk on the infinitely many prices composing the forward curve. More specifically, given a volatility term structure, the drifts of all prices (for each maturity) must satisfy a consistency condition, such that arbitrage opportunities are excluded.
In the classical Heath-Jarrow-Morton framework (HJM in short - see HJM framework on Wikipedia), this condition is expressed in terms of the instantaneous forward rate. Here, we simply want to write this condition in terms of the average forward rate until maturity.

###### Recall : the standard Heath-Jarrow-Morton Framework

Let us assume, as in the standard HJM framework, that the instantaneous forward rate $f(t,T)$ (quoted at time $t$ for a maturity date $T>t$) solves a Stochastic Differential Equation (SDE) :

(1)
\begin{align} df (t,T) = \mu(t,T) dt + \xi(t,T) dW_t. \end{align}

Here $\mu$ and $\xi$ are assumed non-random.

Several stochastic processes (one process for each $T$) are driven by one Brownian motion $W$, and as a consequence, a strong restriction has to be imposed on the drift of these diffusions such that there is no arbitrage opportunity ("no free lunch"), i.e. such that all prices for each maturity become a martingale under one unique change of probability (this change of probability cannot depend on $T$ : in fact we have only one degree of freedom in the variable $t$). The well-known result is the following : the HJM model is arbitrage free if there exists some process $\theta$ not depending on $T$ such that :

(2)
\begin{align} \mu(t,T) = \xi(t,T)\left( \int_t^T \xi(t,s)ds - \theta(t) \right) dt . \end{align}
###### In terms of average forward rate until maturity…

Now let's write the same result in a slightly different form. Consider the average forward rate on the period $[t,T]$, i.e.

(3)
\begin{align} \gamma(t,T) = \frac{\int_t^T f(t,s)ds }{T-t}, \end{align}

and assume

(4)
\begin{align} d\gamma(t,T) = b . dt + \sigma(T-t) dW_t , \end{align}

i.e. we impose a deterministic volatility term structure depending only on the time to maturity $T-t$.

###### What is now the condition on the drift $b$ ensuring that there is no arbitrage opportunity ?

Let us write

(5)
\begin{align} Z(t,T) = (T-t) \gamma(t,T), \end{align}

so

(6)
\begin{align} dZ(t,T) = (T-t) d\gamma(t,T) - \gamma(t,T) dt , \end{align}

which implies

(7)
\begin{align} d\gamma(t,T) = \frac{dZ(t,T)}{T-t} + \frac{\gamma(t,T)}{T-t} dt. \end{align}

On the other hand, since $Z(t,T)= \int_t^T f(t,s)ds$ and $f(t,t)=\gamma(t,t)$,

(8)
\begin{align} dZ(t,T) =\left( -\gamma(t,t) + \int_t^T \mu(t,s)ds \right) dt+ \left( \int_t^T \xi(t,s) ds\right) dW_t, \end{align}

so we must have (by uniqueness of Ito's decomposition) :

(9)
\begin{align} \sigma(T-t) = \frac{ \int_t^T \xi (t,s) ds }{T-t} \mbox{ and } b = \frac{ \int_t^T \mu (t,s) ds + \gamma(t,T) -\gamma(t,t) }{T-t}. \end{align}

So, noting $\sigma'$ the derivative of $\sigma$ with respect to its unique variable, our condition that volatility depends only on the time-to-maturity $\tau=T-t$ yields

(10)
\begin{align} \xi(t,T) = \tau \sigma ' (\tau ) - \sigma(\tau). \end{align}

The no-arbitrage condition now takes the form

(11)
\begin{align} \mu(t,T) = \left( \tau \sigma ' (\tau ) - \sigma(\tau ) \right) \left( \tau \sigma(\tau ) - \theta(t) \right) dt , \end{align}

i.e. we have the following no-arbitrage conditions in terms of average rate :

(12)
\begin{align} b = C(\tau) -\theta(t)\sigma(\tau) -\frac{\gamma(t,T)-\gamma(t,t)}{\tau}, \end{align}

where

(13)
\begin{align} C(\tau) = \frac{1 }{\tau }\int_0^\tau \left( s \sigma' (s) \sigma(s) - \sigma(s)^2 \right) ds. \end{align}
page revision: 35, last edited: 28 Oct 2011 12:36